Erased memory page reconstruction using distributed coding for multiple dimensional parities

ABSTRACT

A method for memory page erasure-correcting property generation in a storage array includes dividing data into multiple stripes for storage in a storage array including multiple storage devices with a topology of a hypercube of a dimension t≥3. The storage devices in same hypercubes of dimension t−1 including the hypercube of dimension t have even parity. An intersection of two non-parallel planes in the hypercube topology is a line, and each point along a line is a storage device in the storage array. Erasure-correcting properties are generated for the data using three nested codes, wherein a first nested code has even parity over planes of class 0, 1 and 2, a second nested code has a first global parity, and a third nested code has a second global parity and a third global parity.

BACKGROUND

Considerable interest has arisen lately in coding schemes that combinelocal and global properties. Applications like Redundant Arrays ofIndependent Disks (RAID) architectures are an example of this interest.In effect, given an array of storage devices, a regular RAIDarchitecture such as RAID 5, protects against a total storage devicefailure. The storage devices may be, for example, solid state devices(SSDs), hard disk drives (HDDs), tapes, or other types of storagedevices. The RAID 5 architecture is implemented by XORing the contentsof the data storage devices, where the outcome of XORing is stored in aparity storage device. Then, if a storage device fails, its contents canbe recovered by XORing the contents of the surviving storage devices.However, RAID 5 may be insufficient for large numbers of storagedevices. More failures may occur exceeding the erasure-correctingcapability of the code. A possible solution is using RAID 6, which addsa second parity, but such an approach may be wasteful.

Other approaches involve using a limited amount of parities, such as oneor two extra parities, called global parities. So, if there is an m×narray of storage devices, each row is protected by a parity like in RAID5. A column of storage devices may represent a box or some otherconfiguration, but it is assumed that a column may fail, in which caseeach element in the column is recovered row by row (locally), like inRAID 5. But if some extra storage devices have failed in addition to thecolumn, data loss will occur if no extra precautions are taken.

SUMMARY

Embodiments relate to storage device arrays including parities inmultiple dimensions for data recovery. In one embodiment, a method formemory page erasure-correcting property generation in a storage arrayincludes dividing data into multiple stripes for storage in a storagearray including multiple storage devices with a topology of a hypercubeof a dimension t≥3. The storage devices in same hypercubes of dimensiont−1 including the hypercube of dimension t have even parity. Anintersection of two non-parallel planes in the hypercube topology is aline, and each point along a line is a storage device in the storagearray. Erasure-correcting properties are generated for the data usingthree nested codes, wherein a first nested code has even parity overplanes of class 0, 1 and 2, a second nested code has a first globalparity, and a third nested code has a second global parity and a thirdglobal parity.

These and other features, aspects and advantages of the embodiments willbecome understood with reference to the following description, appendedclaims and accompanying figures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a cloud computing environment, according to anembodiment;

FIG. 2 depicts a set of abstraction model layers, according to anembodiment;

FIG. 3 illustrates a block diagram of a system for providing parities ina storage array, according to one embodiment;

FIG. 4 illustrates a storage system, according to one embodiment;

FIG. 5 is an example cube diagram illustrating points that correspond tocoordinates, according to an embodiment;

FIG. 6 is another example cube diagram illustrating three-dimensional(3-D) coordinates used for coding, according to an embodiment;

FIG. 7 is a block diagram for an encoding process for separating nodes(SDs) into data SDs and parity SDs, according to an embodiment;

FIG. 8 is a block diagram for a decoding process for the encodingprocess of FIG. 7, according to an embodiment;

FIG. 9 is a block diagram for an example encoding process withadditional global parity, according to an embodiment;

FIG. 10 is a block diagram for an example decoding process for theencoding of FIG. 9, according to an embodiment;

FIG. 11 is a block diagram for a process for additional parities inmultiple dimensions for data recovery, according to an embodiment; and

FIG. 12 illustrates an embodiment of a computer system that may be usedin conjunction with embodiments with additional parities in multipledimensions for data recovery, according to one embodiment.

DETAILED DESCRIPTION

The descriptions of the various embodiments have been presented forpurposes of illustration, but are not intended to be exhaustive orlimited to the embodiments disclosed. Many modifications and variationswill be apparent to those of ordinary skill in the art without departingfrom the scope and spirit of the described embodiments. The terminologyused herein was chosen to best explain the principles of theembodiments, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

It is understood in advance that although this disclosure includes adetailed description of cloud computing, implementation of the teachingsrecited herein are not limited to a cloud computing environment. Rather,embodiments are capable of being implemented in conjunction with anyother type of computing environment now known or later developed.

Cloud computing is a model of service delivery for enabling convenient,on-demand network access to a shared pool of configurable computingresources (e.g., networks, network bandwidth, servers, processing,memory, storage, applications, virtual machines (VMs), and services)that can be rapidly provisioned and released with minimal managementeffort or interaction with a provider of the service. This cloud modelmay include at least five characteristics, at least three servicemodels, and at least four deployment models.

Characteristics are as follows:

On-demand self-service: a cloud consumer can unilaterally provisioncomputing capabilities, such as server time and network storage, asneeded and automatically, without requiring human interaction with theservice's provider.

Broad network access: capabilities are available over a network andaccessed through standard mechanisms that promote use by heterogeneous,thin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

Resource pooling: the provider's computing resources are pooled to servemultiple consumers using a multi-tenant model, with different physicaland virtual resources dynamically assigned and reassigned according todemand. There is a sense of location independence in that the consumergenerally has no control or knowledge over the exact location of theprovided resources but may be able to specify location at a higher levelof abstraction (e.g., country, state, or data center).

Rapid elasticity: capabilities can be rapidly and elasticallyprovisioned and, in some cases, automatically, to quickly scale out andrapidly released to quickly scale in. To the consumer, the capabilitiesavailable for provisioning often appear to be unlimited and can bepurchased in any quantity at any time.

Measured service: cloud systems automatically control and optimizeresource use by leveraging a metering capability at some level ofabstraction appropriate to the type of service (e.g., storage,processing, bandwidth, and active consumer accounts). Resource usage canbe monitored, controlled, and reported, thereby providing transparencyfor both the provider and consumer of the utilized service.

Service Models are as follows:

Software as a Service (SaaS): the capability provided to the consumer isthe ability to use the provider's applications running on a cloudinfrastructure. The applications are accessible from various clientdevices through a thin client interface, such as a web browser (e.g.,web-based email). The consumer does not manage or control the underlyingcloud infrastructure including network, servers, operating systems,storage, or even individual application capabilities, with the possibleexception of limited consumer-specific application configurationsettings.

Platform as a Service (PaaS): the capability provided to the consumer isthe ability to deploy onto the cloud infrastructure consumer-created oracquired applications created using programming languages and toolssupported by the provider. The consumer does not manage or control theunderlying cloud infrastructure including networks, servers, operatingsystems, or storage, but has control over the deployed applications andpossibly application-hosting environment configurations.

Infrastructure as a Service (IaaS): the capability provided to theconsumer is the ability to provision processing, storage, networks, andother fundamental computing resources where the consumer is able todeploy and run arbitrary software, which can include operating systemsand applications. The consumer does not manage or control the underlyingcloud infrastructure but has control over operating systems, storage,deployed applications, and possibly limited control of select networkingcomponents (e.g., host firewalls).

Deployment Models are as follows:

Private cloud: the cloud infrastructure is operated solely for anorganization. It may be managed by the organization or a third party andmay exist on-premises or off-premises.

Community cloud: the cloud infrastructure is shared by severalorganizations and supports a specific community that has shared concerns(e.g., mission, security requirements, policy, and complianceconsiderations). It may be managed by the organizations or a third partyand may exist on-premises or off-premises.

Public cloud: the cloud infrastructure is made available to the generalpublic or a large industry group and is owned by an organization sellingcloud services.

Hybrid cloud: the cloud infrastructure is a composition of two or moreclouds (private, community, or public) that remain unique entities butare bound together by standardized or proprietary technology thatenables data and application portability (e.g., cloud bursting for loadbalancing between clouds).

A cloud computing environment is a service oriented with a focus onstatelessness, low coupling, modularity, and semantic interoperability.At the heart of cloud computing is an infrastructure comprising anetwork of interconnected nodes.

Referring now to FIG. 1, an illustrative cloud computing environment 50is depicted. As shown, cloud computing environment 50 comprises one ormore cloud computing nodes 10 with which local computing devices used bycloud consumers, such as, for example, personal digital assistant (PDA)or cellular telephone 54A, desktop computer 54B, laptop computer 54C,and/or automobile computer system 54N may communicate. Nodes 10 maycommunicate with one another. They may be grouped (not shown) physicallyor virtually, in one or more networks, such as private, community,public, or hybrid clouds as described hereinabove, or a combinationthereof. This allows the cloud computing environment 50 to offerinfrastructure, platforms, and/or software as services for which a cloudconsumer does not need to maintain resources on a local computingdevice. It is understood that the types of computing devices 54A-N shownin FIG. 2 are intended to be illustrative only and that computing nodes10 and cloud computing environment 50 can communicate with any type ofcomputerized device over any type of network and/or network addressableconnection (e.g., using a web browser).

Referring now to FIG. 2, a set of functional abstraction layers providedby the cloud computing environment 50 (FIG. 1) is shown. It should beunderstood in advance that the components, layers, and functions shownin FIG. 2 are intended to be illustrative only and embodiments are notlimited thereto. As depicted, the following layers and correspondingfunctions are provided:

Hardware and software layer 60 includes hardware and softwarecomponents. Examples of hardware components include: mainframes 61; RISC(Reduced Instruction Set Computer) architecture based servers 62;servers 63; blade servers 64; storage devices 65; and networks andnetworking components 66. In some embodiments, software componentsinclude network application server software 67 and database software 68.

Virtualization layer 70 provides an abstraction layer from which thefollowing examples of virtual entities may be provided: virtual servers71; virtual storage 72; virtual networks 73, including virtual privatenetworks; virtual applications and operating systems 74; and virtualclients 75.

In one example, a management layer 80 may provide the functionsdescribed below. Resource provisioning 81 provides dynamic procurementof computing resources and other resources that are utilized to performtasks within the cloud computing environment. Metering and pricing 82provide cost tracking as resources are utilized within the cloudcomputing environment and billing or invoicing for consumption of theseresources. In one example, these resources may comprise applicationsoftware licenses. Security provides identity verification for cloudconsumers and tasks as well as protection for data and other resources.User portal 83 provides access to the cloud computing environment forconsumers and system administrators. Service level management 84provides cloud computing resource allocation and management such thatrequired service levels are met. Service Level Agreement (SLA) planningand fulfillment 85 provide pre-arrangement for, and procurement of,cloud computing resources for which a future requirement is anticipatedin accordance with an SLA.

Workloads layer 90 provides examples of functionality for which thecloud computing environment may be utilized. Examples of workloads andfunctions which may be provided from this layer include: mapping andnavigation 91; software development and lifecycle management 92; virtualclassroom education delivery 93; data analytics processing 94; anderasure-correcting property generation processing 95. As mentionedabove, all of the foregoing examples described with respect to FIG. 2are illustrative only, and the embodiments are not limited to theseexamples.

It is understood all functions of one or more embodiments as describedherein may be typically performed by the system 100 (FIG. 3), system 200(FIG. 4), or the computing system 1200 (FIG. 12), which can be tangiblyembodied as hardware processors and with modules of program code.However, this need not be the case. Rather, the functionality recitedherein could be carried out/implemented and/or enabled by any of thelayers 60, 70, 80 and 90 shown in FIG. 2.

It is reiterated that although this disclosure includes a detaileddescription on cloud computing, implementation of the teachings recitedherein are not limited to a cloud computing environment. Rather, theembodiments may be implemented with any type of clustered computingenvironment now known or later developed.

Flash devices are a type of non-volatile storage devices that can beelectrically erased and reprogrammed in large blocks. Like hard diskdrives (HDDs), flash devices divide the medium into sectors that aretypically 512 bytes long. Flash devices further collect sectors intopages with typically eight sectors per page, so that each page containsfour thousand or 4 kilo (K) bytes. Each sector is protected by an errorcorrecting code (ECC) that corrects a number of single bit errors. Apopular choice, for example, is a BoseChaudhuriHocquenghem (BCH) code,like an eight bit correcting or fifteen bit correcting BCH code,although many variations are possible. BCH codes do hard decisiondecoding. Low Density Parity Check (LDPC) codes are another example of apopular choice, and in this case soft decision is used. As in HDDs,pages in flash devices may suffer hard errors (HEs). This occurs, forexample, when the error correcting capability of the BCH or the LDPCcode is exceeded. As compared to HDDs, exceeding the capability of theerror-correcting code is more likely in flash devices, both as a pagenears the end of its write endurance lifetime, or as a page nears theend of its data retention lifetime. Thus, the number of HEs in flashdevices may be expected to grow over time, leaving latent HEs on adevice. An array made up of flash devices may encounter a mix ofcatastrophic device failures combined with possibly more prevalent HEs.

Addition of first responder parities to data that is stored in a storagearray reduces the number of storage devices that need to be accessed inorder to recover from a single sector erasure in a stripe, as opposed torequiring access to all of the storage devices in the stripe, as is donein a RAID 5 architecture. Reducing the number of devices that areaccessed in the event of a failure may increase the performance of thestorage array. Further, global parities that are used in conjunctionwith row and column parities may also increase the minimum distance ofthe erasure correcting codes to 6, or to 7 or 8 in further embodiments.The minimum distance of the code is a parameter that measures thecorrecting-capability of the code. If the minimum distance of an erasurecorrecting code is d, the code can recover up to d−1 erasures, no matterwhere those erasures are located (for example, multiple erasures in thesame row or column) in the storage array.

The data in the storage array is stored as stripes, in two-dimensional(2-D) storage arrays each stripe comprises an M by N array of symbols,i.e., M rows and N columns. The columns may each correspond torespective individual storage devices of the storage array, which may beflash solid state devices (SSDs). Flash SSDs provide relatively fastmemory; however, over time, SSD memory may degrade as many reads andwrites are performed, leading to errors in which one or more symbols ina stripe are lost. Each symbol may be a fixed-size sector, or page,comprising a fixed number of bytes of data. At least one of the columns(or devices) in a stripe is a row parity column, comprising a column ofsymbols that each give parity information for the specific row in thestripe in which the symbol is located. If a whole device fails, the datathat was stored on the device is recovered by XORing the data stored inthe surviving devices of the storage array. However, additional,isolated pages in other devices may also fail, resulting in data loss.Adding global parity symbols may prevent the data loss. However, all ofthe storage devices may need to be accessed in order to reconstruct thefailed page. If page erasures are common, accessing all of the storagedevices has a negative impact on overall system performance.

FIG. 3 illustrates a block diagram of a system 100 for providingparities in multiple dimensions, according to one embodiment. As shownin FIG. 1, a host computer 102 is in communication with an arraycontroller 104 in a storage system 110. The storage system 110 storesdata in a storage array 108 made up of N storage devices 106 (where N isgreater than one): storage device zero 106A, storage device one 106B,through storage device N−1 106N. In one embodiment, parity bits arestored in the storage devices 106A-N along with host data (e.g.,represented as data bits). In one embodiment, the storage devices 106A-Nin the storage array 108 are implemented by flash devices. In someembodiments, the array 108 is made up of a plurality of flash devices.The system 100 may comprise a RAID 5 or a RAID 6 architecture in variousembodiments. As shown in FIG. 3, the array controller 104 is part of thestorage system 110; in another embodiment, the array controller 104 ispart of the host computer 102.

FIG. 4 illustrates a system 200 including the storage system 110 of FIG.3, according to one embodiment. The storage system 110 may includenumerous other elements such as receivers, transmitters, and clocks asknown by those of ordinary skill in the art, which are not illustratedfor purposes of clarity. As shown in FIG. 4, the array controller 110includes an encoder 202 and a decoder 204. The encoder 202 is utilizedduring a write process for receiving one or more write data pages (e.g.,from the host computer 102) and generating a write stripe, whichincludes both data pages and parity pages. In one embodiment, the writestripe is written in the storage array 108 and spans a plurality ofsegments in the storage array 108. The decoder 204 is utilized during aread process for reading one or more data pages from the storage array108. When one or more hard errors (HEs) in a page are detected, thedecoder reads the whole stripe where the HEs have been detected from thestorage array 108. The decoder 204 and the encoder 202 both have accessto shared data (e.g., data to identify the type of encoding that wasapplied by the encoder 202 to the write pages to generate the writestripe). The read stripe contains parity bits that are removed by thedecoder 204 to generate the read data pages. The decoder 204 includes areconstructor 206 that is used when a read failure of at least one pagehas occurred. A read failure occurs, for example, when the errorcorrecting capability of the internal ECC of a page has been exceeded.Typically, the locations of the pages in error are known and thus, theerror locations (e.g., the erased page location(s)) and the read stripeare sent to the reconstructor 206, which attempts to retrieve the erasedpages. The retrieval of the erased pages, including the process ofcorrecting any errors in the erased pages (using, e.g., an erasurecorrecting code) is referred to herein as reconstructing the erasedpages.

The decoder 204 may reconstruct erased pages as follows. The ECC and/orthe cyclic redundancy check (CRC) in the decoder 204 detect that a pageread has failed. The decoder 204 sends a request to read the pages inthe stripe that are needed to reconstruct the erased page, including theparity data, from storage array 108. The decoder 204 then receives theread pages together with the location(s) of the page(s) where the readhas failed (i.e., the erased page location(s)), and sends the read dataand erasure information to the reconstructor 206. The reconstructor 206then uses the non-erased data, including the parity data, to reconstructthe erased page(s).

In one or more embodiments, instead of 2-D m×n storage device arrays,parities in multiple dimensions are provided (although the example caseof three dimensions (3-D) is described for ease of explanation). In oneor more embodiments, parities in multiple dimensions (e.g., 3-D orhigher dimensions) allow for recovery of a single failure by allowingmultiple paths. Conventionally, the case of 2-D as mentioned above,involves cases with horizontal and vertical parities and extra globalparities.

Consider a code of length n₀n₁n₂ whose coordinates are labeled by a 3-Darray (that is referred herein as an (n₀, n₁, n₂)-cube) (i, j,

), where where 0≤i≤n₀−1, 0≤j≤n₁−1 and 0≤

≤n₂ ⁻¹, an n is a positive integer. The elements of the code are denotedby

, and when written in one dimension, the following correspondence isused for the one-dimensional ordering of the coordinates:

(i,j,

)↔in₁ n ₂ +jn ₂+

.  (EQ. 8)

In one or more embodiments, there are n₀ parallel planes of n₁n₂elements each, that is, for each 0≤i≤n₀−1, the planes consist of then₁n₂ entries (i, j,

) such that 0≤j≤n₁−1 and 0≤

≤n₂−1. These n₀ planes are referred to as planes of class 0. Similarly,there are n₁ planes of class 1 (having n₀n₂ elements each), and n₂planes of class 2 (having n₀n₁ elements each).

In one embodiment, a line is referred to as the intersection of twoplanes. For example, given i and j, where 0≤i≤n₀−1 and 0≤j≤n₁−1, the n₂coordinates (i, j,

), where 0≤

≤n₂−1, constitute a line of length n₂. In one embodiment, the encoder202 constructs three nested (i.e., contained within each other) codesC_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽²⁾⊂C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽¹⁾⊂C_(n) ₀ _(,n) ₁_(,n) ₂ ⁽⁰⁾ of length n₀n₁n₂. C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽⁰⁾ is a code witheven parity over each of the n₀+n₁+n₂ planes of class 0, 1 and 2, whileC_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽¹⁾ and C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽²⁾ have one andtwo extra (global) parities respectively.

Consider a finite field GF(2^(b)). Denote by I_(j) the j×j identitymatrix, by e _(j) the all-1 vector of length j and by ⊗ the tensorproduct of two matrices. The parity-check matrix of code C_(n) ₀ _(,n) ₁_(,n) ₂ ⁽⁰⁾ is given by then (n₀+n₁+n₂)×n₀n₁n₂ matrix

$\begin{matrix}{H_{n_{0},n_{1},n_{2}}^{(0)} = {\begin{pmatrix}{I_{n_{0}} \otimes {\underset{\_}{e}}_{n_{1}n_{2}}} \\{{\underset{\_}{e}}_{n_{0}} \otimes \left( {I_{n_{2}} \otimes {\underset{\_}{e}}_{n_{1}}} \right)} \\{{\underset{\_}{e}}_{n_{0}} \otimes \left( {{\underset{\_}{e}}_{n_{2}} \otimes I_{n_{1}}} \right)}\end{pmatrix}.}} & \left( {{EQ}.\mspace{14mu} 9} \right)\end{matrix}$

It can easily be seen that there are two linearly dependent rows H_(n) ₀_(,n) ₁ _(,n) ₂ ⁽⁰⁾ as given by (EQ. 9): the last row in e _(n) ₀⊗(I_(n) ₂ ⊗e _(n) ₁ ) and the last row in e _(n) ₀ ⊗(e _(n) ₂ ⊗I_(n) ₁ )can be eliminated, since XORing all the rows of

$\begin{pmatrix}{I_{n_{0}} \otimes {\underset{\_}{e}}_{n_{1}n_{2}}} \\{{\underset{\_}{e}}_{n_{0}} \otimes \left( {I_{n_{2}} \otimes {\underset{\_}{e}}_{n_{1}}} \right)}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {of}\mspace{14mu} \begin{pmatrix}{I_{n_{0}} \otimes {\underset{\_}{e}}_{n_{1}n_{2}}} \\{{\underset{\_}{e}}_{n_{0}} \otimes \left( {{\underset{\_}{e}}_{n_{2}} \otimes I_{n_{1}}} \right)}\end{pmatrix}$

gives the zero vector. The remaining n₀+n₁+n₂−2 rows of H_(n) ₀ _(,n) ₁_(,n) ₂ ⁽⁰⁾ are linearly independent, so C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽⁰⁾ hasdimension n₀n₁n₂−n₀−n₁−n₂−n₂+2. Thus, C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽⁰⁾ is an[n₀n₁n₂,n₀n₁n₂−n₀−n₁−n₂+2] code.

C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽¹⁾ is defined as the [n₀n₁n₂,n₀n₁n₂−n₀−n₁−n₂+1]code whose parity-check matrix is given by

$\begin{matrix}{{H_{n_{0},n_{1},n_{2}}^{(1)} = \begin{pmatrix}H_{n_{0},n_{1},n_{2}}^{(0)} \\{1\mspace{14mu} \alpha \mspace{14mu} \alpha^{2}\mspace{14mu} \ldots \mspace{14mu} \alpha^{{n_{0}n_{1}n_{2}} - 1}}\end{pmatrix}},} & \left( {{EQ}.\mspace{14mu} 10} \right)\end{matrix}$

and C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽²⁾ as the [n₀n₁n₂,n₀n₁n₂−n₀−n₁−n₂−2] codewhose parity-check matrix is given by

$\begin{matrix}{{H_{n_{0},n_{1},n_{2}}^{(2)} = \begin{pmatrix}H_{n_{0},n_{1},n_{2}}^{(0)} \\{1\mspace{14mu} \alpha \mspace{14mu} \alpha^{2}\mspace{14mu} \ldots \mspace{14mu} \alpha^{{n_{0}n_{1}n_{2}} - 1}} \\{1\mspace{14mu} \alpha^{- 1}\mspace{14mu} \alpha^{- 2}\mspace{14mu} \ldots \mspace{14mu} \alpha^{{{- n_{0}}n_{1}n_{2}} + 1}}\end{pmatrix}},} & \left( {{EQ}.\mspace{14mu} 11} \right)\end{matrix}$

where H_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽⁰⁾ is given by (EQ 9).

The constructions are illustrated with an example where FIGS. 5-10assist in understanding one or more embodiments. Consider codesC_(2,3,3) ⁽⁰⁾, C_(2,3,3) ⁽¹⁾ and C_(2,3,3) ⁽²⁾. In one embodiment, thecoordinates correspond to a 2×3×3 cube (see, e.g., FIGS. 5-6). The cubeconsists of 2 planes of class 0 containing 9 elements each, 3 planes ofclass 1 containing 6 elements each, and 3 planes of class 2 alsocontaining 6 elements each, where each plane corresponds to a localparity (two of these local parities are dependent, so there are 6independent local parities). In addition, one or two global parities areadded illustrating the three codes in one or more embodiments.

Consider the finite field GF(32) and let α be a primitive element inGF(32) in one example embodiment. According to (EQ. 9), (EQ. 10) and(EQ. 11), the parity-check matrices of C_(2,3,3) ⁽⁰⁾, C_(2,3,3) ⁽¹⁾ andC_(2,3,3) ⁽²⁾ are given by

$\begin{matrix}{H_{2,3,3}^{(0)} = \begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1\end{pmatrix}} \\{H_{2,3,3}^{(1)} = \begin{pmatrix}H_{2,3,3}^{(0)} \\{1\mspace{14mu} \alpha \mspace{14mu} \alpha^{2}\mspace{14mu} \ldots \mspace{14mu} \alpha^{17}}\end{pmatrix}} \\{H_{2,3,3}^{(2)} = \begin{pmatrix}H_{2,3,3}^{(0)} \\{1\mspace{14mu} \alpha \mspace{14mu} \alpha^{2}\mspace{14mu} \ldots \mspace{14mu} \alpha^{17}} \\{1\mspace{14mu} \alpha^{30}\mspace{14mu} \alpha^{29}\mspace{14mu} \ldots \mspace{14mu} \alpha^{14}}\end{pmatrix}}\end{matrix}$

respectively. The fifth and eighth rows from each of these matrices maybe eliminated since they are linearly dependent.

In one embodiment, first the erasure-correcting properties of C_(n) ₀_(,n) ₁ _(,n) ₂ ⁽⁰⁾ is provided in the following. Code C_(n) ₀ _(,n) ₁_(,n) ₂ ⁽⁰⁾ can correct any erased line and has minimum distance 4.Without loss of generality, consider the line given by i₀ and j₀, where0≤i₀≤n₀−1 and 0≤j0≤n₁−1, and assume that the elements on that line havebeen erased. There are n₂ parallel planes through each point (i₀, j₀,

), 0≤

≤n₂−1 in the line. Since there is even parity on each of the planes,and, except for the erased element on the line, no other element hasbeen erased, the erased element may be retrieved by XORing thenon-erased elements in the plane.

In one embodiment, it can be seen that any three erased elements may becorrected. If the three elements are in the same line, then they arerecovered as provided above. If one of the elements is not in a linewith the other two, there is certainly a plane through it that does notcontain any of the other two elements. This non-aligned erasure can becorrected using the parity on that plane. This leaves two erasures thatare corrected similarly.

In one example embodiment, the code has codewords of weight 4. Thesecorrespond to codewords whose non-zero elements are in the vertices ofrectangles in a plane. For example, consider coordinates (0,0,0),(0,0,1), (0,1,0) and (0,1,1) shown in FIG. 6. According to therelationship given by (EQ. 8), these coordinates, when taken in onedimension, correspond to 0, 1, n₂ and n₂+1 respectively. Assume that avector is 1 in these four coordinates and 0 elsewhere. In oneembodiment, it can be verified that the syndrome of such a vector withrespect to parity-check matrix H_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽⁰⁾ as given by(EQ. 9) is zero, which is a codeword of weight 4.

Next the special case n₀=n₁=n₂=n is considered. Codes C_(n,n,n) ^((i))are denoted as C_(n) ^((i)), where 0≤i≤2. Similarly, denoted by H_(n)^((i)) are the parity check matrices corresponding to codes C_(n)^((i)). In one embodiment, minimum distance properties may be proved forC_(n) ⁽¹⁾ and C_(n) ⁽²⁾, provided a finite field of size large enough isused. In one example embodiment, consider code C_(n) ⁽¹⁾ over the fieldGF(2^(b)), and assume that n³≤2^(b)−1. Then C_(n) ⁽¹⁾ has minimumdistance 6 and any five erasures can be corrected.

In one embodiment, consider code C_(n) ⁽²⁾ over the field GF(2^(b)),where

n(n−1)≤2^(b)−1

Then C_(n) ⁽²⁾ has minimum distance d≥7 and any six erasures can becorrected.

Although the description involves cubes and single parity over thedifferent planes composing the cubes, the construction of codes in oneor more embodiments is extended to hypercubes of dimension t>3, wherethe single parity will occur now over the hypercubes of dimension t−1composing the hypercube of dimension t.

FIG. 5 is an example cube 500 diagram illustrating points thatcorrespond to coordinates, according to an embodiment. In the examplecube 500, the 18 coordinates of the code correspond to points in thecube 500. In the example 500, these are numerated from 0 to 17. For the3-D example described above, there are two parallel planes of type 0,each one containing 9 points: planes {0,1,2,3,4,5,6,7,8} and{9,10,11,12,13,14,15,16,17}, which is referred to as planes of type 0.There are three parallel planes of type 1, each one containing 6 points:{0,3,6,9,12,15}, {1,4,7,10,13,16} and {2,5,8,11,14,17}. There are alsothree parallel planes of type 2, each one containing 6 points:{0,1,2,9,10,11}, {3,4,5,11,13,14} and {6,7,8,15,16,17}.

In one example embodiment, it can be seen that the intersection of twonon-parallel planes provides a line. For example, the intersection ofplane {0,1,2,3,4,5,6,7,8} with plane {1,4,7,10,13,16} provides line{1,4,7}. In one embodiment, each point is, for example, a storage device(SD). Each plane corresponds to even parity. In one example embodiment,if SD 7 fails, it can be recovered using plane {0,1,2,3,4,5,6,7,8}: theXOR of SDs {0,1,2,3,4,5,6,8} provides the contents of the failed SD 7.Alternatively, in one embodiment it can be recovered using plane{1,4,7,10,13,16} or plane {6,7,8,15,16,17}. Therefore, in one embodimentthere are three alternative ways to recover the failed SD.

In one embodiment, a whole line may be recovered, such as theaforementioned {1,4,7} line: in this case, SD 1 is recovered using plane{0,1,2,9,10,11} (notice that the only failed SD is SD 1 in this plane),SD 4 using plane {3,4,5,12,13,14} and SD 7 using plane {6,7,8,15,16,17}.In one embodiment, if the parities on planes are only used, this is whatis referred to as code C⁽⁰⁾ _(2,3,3). But this code has limitations. Inone example, rectangles, such as {0,1,6,7} cannot be recovered. To thisend, in one embodiment a global parity is required, such as the globalparity provided by code C⁽¹⁾ _(2,3,3), which allows to correctrectangles or any five failed SDs for that matter.

FIG. 6 is another example cube 600 diagram illustrating 3-D coordinatesused for coding, according to an embodiment. Cube 600 is similar to cube500. Cube 500 (FIG. 5) and 600, together, assist in illustrating theassignment of EQ. 8 described above.

FIG. 7 is a block diagram for an encoding process 700 for separatingnodes (e.g., SDs) into data SDs and parity SDs, according to anembodiment. Process 700 shows explicitly how to obtain the parity nodesfrom the data nodes. The process 700 is provided to assist with theexample embodiment described above for a 3-D storage array. It should benoted that the general case proceeds similarly. The decoding of codeC⁽⁰⁾ _(2,3,3) whose encoding process is given by 700 is provided in FIG.8. In one example embodiment, process 700 begins with block 710 wherethe division of the SDs occur. As shown, the SDs are separated into dataSDs and Parity SDs.

In one embodiment, in block 720, SD 1 is obtained by performing XORingon SDs 4, 7, 10, 13 and 16. In one embodiment, in block 730, SD 2 isobtained by performing XORing on SDs 5, 8, 11, 14 and 17. In oneembodiment, in block 740, SD 3 is obtained by performing XORing on SDs4, 5, 12, 13 and 14. In one embodiment, in block 750, SD 6 is obtainedby performing XORing on SDs 7, 8, 15, 16 and 17. In one embodiment, inblock 760, SD 0 is obtained by performing XORing on SDs 1, 2, 3, 4, 5,6, 7 and 8. In one embodiment, in block 770, SD 9 is obtained byperforming XORing on SDs 0, 3, 6, 12 and 15.

FIG. 8 is a block diagram for a decoding process 800 for the encodingprocess of FIG. 7, according to an embodiment. In one embodiment, inprocess 800 particular parameters are not taken, instead the process 800proceeds in general. In one embodiment, it should be noted that theencoding (e.g., process 700, FIG. 7) is a special case of the decodingprocess 800, which is always true for error-correcting codes (the paritySDs may be considered as failed SDs and are recovered from the survivingones). In one embodiment, the decoding process 800 consists ofiterations over all possible planes until all the failed SDs arecorrected, or there is an iteration providing no new corrections andfailed SDs remain. In that case it may be declared that there exists anuncorrectable error.

In one embodiment, in block 810 for each plane, if there are no failedSDs or there are more than one failed SD, the process 800 does nothingand proceeds to block 820. In block 820, process 800 corrects the failedSD by XORing the remaining SDs in the plane. In one embodiment, in block830 the process 800 is repeated until all failed SDs are recovered. Ifafter an iteration there are no new SDs that are recovered, but failedSDs remain, an uncorrectable pattern is declared.

FIG. 9 is a block diagram for an example encoding process 900 withadditional global parity, according to an embodiment. The encodingprocess 900 assists with the example described above for the 3-D storagedevice array. In one example embodiment, process 900 is similar toprocess 700 (FIG. 7) but with an extra (global) parity. While theprocesses 700 and 900 are similar, at the end, process 900 requiressolving 4 parity SDs simultaneously, which can be done by using theglobal parity.

In one embodiment, in block 910 the SDs are divided into data SDs andparity SDs. In block 920, SD2 is obtained by XORing SDs 5, 8, 11, 14 and17. In block 930, SD6 is obtained by XORing SDs 7, 8, 15, 16 and 17. Inblock 940, SD9 is obtained by XORing SDs 10, 11, 12, 13, 14, 15, 16 and17. In one embodiment, in block 950 SDs 0, 1, 3 and 4 are obtained bysolving a system of four linear equations with four unknowns utilizingthe global parity in addition to three local parities.

FIG. 10 is a block diagram for an example decoding process 1000 for theencoding of FIG. 9, according to an embodiment. In one embodiment,process 1000 illustrates the decoding of process 900 (FIG. 9) ingeneral. In one embodiment, at the end of process 1000, if decodingfails, it attempts to correct the remaining failed SDs provided thatthere are no more than four failed SDs.

In one embodiment, in block 1010 all possible failed SDs are correctedusing the decoding for C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽⁰⁾. In block 1020, ifmore than four failed SDs remain, an uncorrectable pattern is declared.In one embodiment, in block 1030, the four failed SDs are corrected bysolving a linear system of four equations with four unknowns using thelocal parities and the global parity.

FIG. 11 illustrates a block diagram for a process 1100 for distributedcoding, according to one embodiment. In one embodiment, in block 1110,process 1000 includes dividing data into multiple stripes for storage ina storage array comprising storage devices with a topology of ahypercube of a dimension t≤3. The storage devices in the same hypercubesof dimension t−1 comprising the hypercube of a dimension t have evenparity.

In one embodiment, in process 1100 the storage devices are linked by oneor more global parities over a finite field. In one embodiment, inprocess 1100 the storage devices are linked by one global parity, andthe minimum distance of a code is 6. In one embodiment, the storagedevices are linked by two global parities, and the minimum distance of acode is at least 7.

In one embodiment, in process 1100 a line is an intersection of any twoplanes of the multiple planes. In one embodiment, process 1100 mayfurther include retrieving any erased elements of a line by XORingnon-erased elements in a plane. In one embodiment, process 1100 mayfurther include correcting a number of erasures using the one or moreglobal parities. IN one embodiment, in block 1120 one or more globalparities are added to the hypercube such that a minimum distance of acode is enhanced.

FIG. 12 illustrates an example of a computer 1200 which may be utilizedby exemplary embodiments of first responder parity for a storage array.Various operations discussed above may utilize the capabilities of thecomputer 1200. One or more of the capabilities of the computer 1200 maybe incorporated in any element, module, application, and/or componentdiscussed herein.

The computer 1200 includes, but is not limited to, PCs, workstations,laptops, PDAs, palm devices, servers, storages, and the like. Generally,in terms of hardware architecture, the computer 1200 may include one ormore processors 1210, memory 1220, and one or more I/O devices 1270 thatare communicatively coupled via a local interface (not shown). The localinterface can be, for example but not limited to, one or more buses orother wired or wireless connections, as is known in the art. The localinterface may have additional elements, such as controllers, buffers(caches), drivers, repeaters, and receivers, to enable communications.Further, the local interface may include address, control, and/or dataconnections to enable appropriate communications among theaforementioned components.

The processor 1210 is a hardware device for executing software that canbe stored in the memory 1220. The processor 1210 can be virtually anycustom made or commercially available processor, a central processingunit (CPU), a digital signal processor (DSP), or an auxiliary processoramong several processors associated with the computer 1200, and theprocessor 1210 may be a semiconductor based microprocessor (in the formof a microchip) or a macroprocessor.

The memory 1220 can include any one or combination of volatile memoryelements (e.g., random access memory (RAM), such as dynamic randomaccess memory (DRAM), static random access memory (SRAM), etc.) andnonvolatile memory elements (e.g., ROM, erasable programmable read onlymemory (EPROM), electronically erasable programmable read only memory(EEPROM), programmable read only memory (PROM), tape, compact disc readonly memory (CD-ROM), disk, diskette, cartridge, cassette or the like,etc.). Moreover, the memory 1220 may incorporate electronic, magnetic,optical, and/or other types of storage media. Note that the memory 1220can have a distributed architecture, where various components aresituated remote from one another, but can be accessed by the processor1210.

The software in the memory 1220 may include one or more separateprograms, each of which comprises an ordered listing of executableinstructions for implementing logical functions. The software in thememory 1220 includes a suitable operating system (O/S) 1250, compiler1240, source code 1230, and one or more applications 1260 in accordancewith exemplary embodiments. As illustrated, the application 1260comprises numerous functional components for implementing the featuresand operations of the exemplary embodiments. The application 1260 of thecomputer 1200 may represent various applications, computational units,logic, functional units, processes, operations, virtual entities, and/ormodules in accordance with exemplary embodiments, but the application1260 is not meant to be a limitation.

The operating system 1250 controls the execution of other computerprograms, and provides scheduling, input-output control, file and datamanagement, memory management, and communication control and relatedservices. It is contemplated by the inventors that the application 1260for implementing exemplary embodiments may be applicable on allcommercially available operating systems.

Application 1260 may be a source program, executable program (objectcode), script, or any other entity comprising a set of instructions tobe performed. When a source program, then the program is usuallytranslated via a compiler (such as the compiler 1240), assembler,interpreter, or the like, which may or may not be included within thememory 1220, so as to operate properly in connection with the O/S 1250.Furthermore, the application 1260 can be written as an object orientedprogramming language, which has classes of data and methods, or aprocedure programming language, which has routines, subroutines, and/orfunctions, for example but not limited to, C, C++, C#, Pascal, BASIC,API calls, HTML, XHTML, XML, ASP scripts, FORTRAN, COBOL, Perl, Java,ADA, .NET, and the like.

The I/O devices 1270 may include input devices such as, for example butnot limited to, a mouse, keyboard, scanner, microphone, camera, etc.Furthermore, the I/O devices 1270 may also include output devices, forexample but not limited to a printer, display, etc. Finally, the I/Odevices 1270 may further include devices that communicate both inputsand outputs, for instance but not limited to, a NIC ormodulator/demodulator (for accessing remote devices, other files,devices, systems, or a network), a radio frequency (RF) or othertransceiver, a telephonic interface, a bridge, a router, etc. The I/Odevices 1270 also include components for communicating over variousnetworks, such as the Internet or intranet.

If the computer 1200 is a PC, workstation, intelligent device or thelike, the software in the memory 1220 may further include a basic inputoutput system (BIOS) (omitted for simplicity). The BIOS is a set ofessential software routines that initialize and test hardware atstartup, start the O/S 1250, and support the transfer of data among thehardware devices. The BIOS is stored in some type of read-only-memory,such as ROM, PROM, EPROM, EEPROM or the like, so that the BIOS can beexecuted when the computer 1200 is activated.

When the computer 1200 is in operation, the processor 1210 is configuredto execute software stored within the memory 1220, to communicate datato and from the memory 1220, and to generally control operations of thecomputer 1200 pursuant to the software. The application 1260 and the O/S1250 are read, in whole or in part, by the processor 1210, perhapsbuffered within the processor 1210, and then executed.

When the application 1260 is implemented in software it should be notedthat the application 1260 can be stored on virtually any computerreadable storage medium for use by or in connection with any computerrelated system or method. In the context of this document, a computerreadable storage medium may be an electronic, magnetic, optical, orother physical device or means that can contain or store a computerprogram for use by or in connection with a computer related system ormethod.

The application 1260 can be embodied in any computer-readable storagemedium for use by or in connection with an instruction execution system,apparatus, or device, such as a computer-based system,processor-containing system, or other system that can fetch theinstructions from the instruction execution system, apparatus, or deviceand execute the instructions. In the context of this document, a“computer-readable storage medium” can be any means that can store theprogram for use by or in connection with the instruction executionsystem, apparatus, or device. The computer readable storage medium canbe, for example but not limited to, an electronic, magnetic, optical,electromagnetic, or semiconductor system, apparatus, or a device.

More specific examples (a nonexhaustive list) of the computer-readablestorage medium may include the following: an electrical connection(electronic) having one or more wires, a portable computer diskette(magnetic or optical), a random access memory (RAM) (electronic), aread-only memory (ROM) (electronic), an erasable programmable read-onlymemory (EPROM, EEPROM, or Flash memory) (electronic), an optical fiber(optical), and a portable compact disc memory (CDROM, CD R/W) (optical).Note that the computer-readable storage medium could even be paper oranother suitable medium, upon which the program is printed or punched,as the program can be electronically captured, via for instance opticalscanning of the paper or other medium, then compiled, interpreted orotherwise processed in a suitable manner if necessary, and then storedin a computer memory.

In exemplary embodiments, where the application 1260 is implemented inhardware, the application 1260 can be implemented with any one or acombination of the following technologies, which are well known in theart: a discrete logic circuit(s) having logic gates for implementinglogic functions upon data signals, an application specific integratedcircuit (ASIC) having appropriate combinational logic gates, aprogrammable gate array(s) (PGA), a field programmable gate array(FPGA), etc.

In one or more embodiments, technical effects and benefits includecorrection of errors in a storage array by reading from a reduced numberof storage devices of the storage array.

As will be appreciated by one skilled in the art, aspects of theembodiments may be embodied as a system, method or computer programproduct. Accordingly, aspects of the embodiments may take the form of anentirely hardware embodiment, an entirely software embodiment (includingfirmware, resident software, micro-code, etc.) or an embodimentcombining software and hardware aspects that may all generally bereferred to herein as a “circuit,” “module” or “system.” Furthermore,aspects of the embodiments may take the form of a computer programproduct embodied in one or more computer readable medium(s) havingcomputer readable program code embodied thereon.

Any combination of one or more computer readable medium(s) may beutilized. The computer readable medium may be a computer readable signalmedium or a computer readable storage medium. A computer readablestorage medium may be, for example, but not limited to, an electronic,magnetic, optical, electromagnetic, infrared, or semiconductor system,apparatus, or device, or any suitable combination of the foregoing. Morespecific examples (a nonexhaustive list) of the computer readablestorage medium would include the following: an electrical connectionhaving one or more wires, a portable computer diskette, a hard disk, arandom access memory (RAM), a read-only memory (ROM), an erasableprogrammable read-only memory (EPROM or Flash memory), an optical fiber,a portable compact disc read-only memory (CD-ROM), an optical storagedevice, a magnetic storage device, or any suitable combination of theforegoing. In the context of this document, a computer readable storagemedium may be any tangible medium that can contain, or store a programfor use by or in connection with an instruction execution system,apparatus, or device.

A computer readable signal medium may include a propagated data signalwith computer readable program code embodied therein, for example, inbaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including, but not limited to,electro-magnetic, optical, or any suitable combination thereof. Acomputer readable signal medium may be any computer readable medium thatis not a computer readable storage medium and that can communicate,propagate, or transport a program for use by or in connection with aninstruction execution system, apparatus, or device.

Program code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wireline, optical fiber cable, RF, etc., or any suitable combination ofthe foregoing.

Computer program code for carrying out operations for aspects of theembodiments may be written in any combination of one or more programminglanguages, including an object oriented programming language such asJava, Smalltalk, C++ or the like and conventional procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The program code may execute entirely on the user's computer,partly on the user's computer, as a stand-alone software package, partlyon the user's computer and partly on a remote computer or entirely onthe remote computer or server. In the latter scenario, the remotecomputer may be connected to the user's computer through any type ofnetwork, including a local area network (LAN) or a wide area network(WAN), or the connection may be made to an external computer (forexample, through the Internet using an Internet Service Provider).

Aspects of the embodiments are described below with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems) and computer program products according to the embodiments. Itwill be understood that each block of the flowchart illustrations and/orblock diagrams, and combinations of blocks in the flowchartillustrations and/or block diagrams, can be implemented by computerprogram instructions. These computer program instructions may beprovided to a processor of a general purpose computer, special purposecomputer, or other programmable data processing apparatus to produce amachine, such that the instructions, which execute via the processor ofthe computer or other programmable data processing apparatus, createmeans for implementing the functions/acts specified in the flowchartand/or block diagram block or blocks.

These computer program instructions may also be stored in a computerreadable medium that can direct a computer, other programmable dataprocessing apparatus, or other devices to function in a particularmanner, such that the instructions stored in the computer readablemedium produce an article of manufacture including instructions whichimplement the function/act specified in the flowchart and/or blockdiagram block or blocks.

The computer program instructions may also be loaded onto a computer,other programmable data processing apparatus, or other devices to causea series of operational steps to be performed on the computer, otherprogrammable apparatus or other devices to produce a computerimplemented process such that the instructions which execute on thecomputer or other programmable apparatus provide processes forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments. In this regard, each block in the flowchart or blockdiagrams may represent a module, segment, or portion of instructions,which comprises one or more executable instructions for implementing thespecified logical function(s). In some alternative implementations, thefunctions noted in the block may occur out of the order noted in thefigures. For example, two blocks shown in succession may, in fact, beexecuted substantially concurrently, or the blocks may sometimes beexecuted in the reverse order, depending upon the functionalityinvolved. It will also be noted that each block of the block diagramsand/or flowchart illustration, and combinations of blocks in the blockdiagrams and/or flowchart illustration, can be implemented by specialpurpose hardware-based systems that perform the specified functions oracts or carry out combinations of special purpose hardware and computerinstructions.

References in the claims to an element in the singular is not intendedto mean “one and only” unless explicitly so stated, but rather “one ormore.” All structural and functional equivalents to the elements of theabove-described exemplary embodiment that are currently known or latercome to be known to those of ordinary skill in the art are intended tobe encompassed by the present claims. No claim element herein is to beconstrued under the provisions of 35 U.S.C. section 112, sixthparagraph, unless the element is expressly recited using the phrase“means for” or “step for.”

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the embodiments.As used herein, the singular forms “a”, “an” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “comprises”and/or “comprising,” when used in this specification, specify thepresence of stated features, integers, steps, operations, elements,and/or components, but do not preclude the presence or addition of oneor more other features, integers, steps, operations, elements,components, and/or groups thereof.

The corresponding structures, materials, acts, and equivalents of allmeans or step plus function elements in the claims below are intended toinclude any structure, material, or act for performing the function incombination with other claimed elements as specifically claimed. Thedescription of the embodiments has been presented for purposes ofillustration and description, but is not intended to be exhaustive orlimited to the embodiments in the form disclosed. Many modifications andvariations will be apparent to those of ordinary skill in the artwithout departing from the scope and spirit of the embodiments. Theembodiments were chosen and described in order to best explain theprinciples of the embodiments and the practical application, and toenable others of ordinary skill in the art to understand the variousembodiments with various modifications as are suited to the particularuse contemplated.

What is claimed is:
 1. A method for memory page erasure-correctingproperty generation in a storage array comprising: dividing data into aplurality of stripes for storage in a storage array comprising aplurality of storage devices with a topology of a hypercube of adimension t≥3, wherein the storage devices in same hypercubes ofdimension t−1 comprising the hypercube of dimension t have even parity,an intersection of two non-parallel planes in the hypercube topology isa line, and each point along a line is a storage device in the storagearray; and generating erasure-correcting properties for the data usingthree nested codes, wherein a first nested code has even parity overplanes of class 0, 1 and 2, a second nested code has a first globalparity, and a third nested code has a second global parity and a thirdglobal parity.
 2. The method of claim 1, further comprising:reconstructing, by a reconstructor processor, erased data for erasedmemory pages from non-erased data in the storage array by using paritiesin at least three dimensions based on the hypercube topology of thestorage devices.
 3. The method of claim 2, wherein storage deviceaccesses in the storage array are limited for reconstruction to at leastone plane within the hypercube topology including non-erased memorypages, and a minimum distance d for the three nested codes providesreconstruction of d−1 erasures in the storage array.
 4. The method ofclaim 3, wherein: the at least one plane comprises at least one of: afirst plane including the erased data, and any other plane thatintersects with a particular storage device in the hypercube topologythat included the erased data; and the storage devices are linked by oneor more of the first global parity, the second global parity and thethird global parity over a finite field.
 5. The method of claim 2,wherein the storage devices are linked by one of the first globalparity, the second global parity and the third global parity, and d is6.
 6. The method of claim 2, wherein the storage devices are linked bytwo of: the first global parity, the second global parity and the thirdglobal parity, and d is at least
 7. 7. The method of claim 2, furthercomprising: retrieving any erased elements of a line by XORingnon-erased elements in a plane; and correcting a number of erasuresusing one or more of the first global parity, the second global parityand the third global parity.
 8. A computer program product for memorypage erasure-correcting property generation in a storage array, thecomputer program product comprising a computer readable storage mediumhaving program instructions embodied therewith, the program instructionsexecutable by a processor to cause the processor to: divide, by theprocessor, data into a plurality of stripes for storage in a storagearray comprising a plurality of storage devices with a topology of ahypercube of a dimension t≤3, wherein the storage devices in samehypercubes of dimension t−1 comprising the hypercube of dimension t haveeven parity, an intersection of two non-parallel planes in the hypercubetopology is a line, and each point along a line is a storage device inthe storage array; and generate, by the processor, erasure-correctingproperties for the data using three nested codes, wherein a first nestedcode has even parity over planes of class 0, 1 and 2, a second nestedcode has a first global parity, and a third nested code has a secondglobal parity and a third global parity.
 9. The computer program productof claim 8, further comprising program instructions executable by theprocessor to cause the processor to: reconstruct, by the processor,erased data for erased memory pages from non-erased data in the storagearray by using parities in at least three dimensions based on thehypercube topology of the storage devices.
 10. The computer programproduct of claim 9, wherein storage device accesses are limited forreconstruction to at least one plane within the hypercube topologyincluding non-erased memory pages, and a minimum distance d for thethree nested codes provides reconstruction of d−1 erasures in thestorage array.
 11. The computer program product of claim 10, wherein:the storage devices are linked by one or more of the first globalparity, the second global parity and the third global parity over afinite field; the at least one plane within the hypercube topologycomprises at least one of: a first plane including the erased data; andany plane that intersects with a particular storage device in thehypercube topology that included the erased data.
 12. The computerprogram product of claim 9, wherein the storage devices are linked byone of the first global parity, the second global parity and the thirdglobal parity, and d is
 6. 13. The computer program product of claim 9,wherein the storage devices are linked by two of: the first globalparity, the second global parity and the third global parity, and d isat least
 7. 14. The computer program product of claim 9, wherein: t=3;wherein: the hypercube topology comprises n₀ parallel planes of n₁n₂elements each, such that each 0≤i≤n₀−1; planes in the hypercube topologycomprise the n₁n₂ elements entries (i, j,

) such that 0≤j≤n₁−1 and 0≤

≤n₂−1, where n₀ planes are class 0, n₁ planes are class 1 including n₀n₂elements each, and n₂ planes are class 2 including n₀n₁ elements each;and the three nested codes are represented as: C_(n) ₀ _(,n) ₁ _(,n) ₂⁽²⁾⊂C_(n) ₀ _(,n) ₁ _(,n) ₂ ₍₁₎⊂C_(n) ₀ _(,n) ₁ _(,n) ₂ ⁽⁰⁾ of lengthn₀n₁n₂.
 15. An array of storage devices comprising: a storage devicearray comprising a plurality of storage devices with a topology of ahypercube of a dimension t≤3, wherein the plurality of storage devicesin same hypercubes of dimension t−1 comprising the hypercube of adimension/have even parity, an intersection of two non-parallel planesin the hypercube topology is a line, and each point along a line is astorage device in the storage array; and a processor configured to:divide data into a plurality of stripes for storage in the storagearray; and generate erasure-correcting properties for the data usingthree nested codes, wherein a first nested code has even parity overplanes of class 0, 1 and 2, a second nested code has a first globalparity, and a third nested code has a second global parity and a thirdglobal parity.
 16. The array of storage devices of claim 15, wherein theprocessor is further configured to: reconstruct erased data for erasedmemory pages from non-erased data in the storage array by using paritiesin at least three dimensions based on the hypercube topology of thestorage devices.
 17. The array of storage devices of claim 16, whereinthe processor is further configured to: limit storage device accesses inthe storage array for reconstruction to at least one plane within thehypercube topology including non-erased memory pages, wherein a minimumdistance d for the three nested codes provides reconstruction of d−1erasures in the storage array.
 18. The array of storage devices of claim17, wherein: the at least one plane within the hypercube topologycomprises at least one of: a first plane including the erased data, andany other plane that intersects with a particular storage device in thehypercube topology that included the erased data; and the plurality ofstorage devices are linked by one or more of the first global parity,the second global parity and the third global parity over a finitefield.
 19. The array of storage devices of claim 16, wherein theplurality of storage devices are linked by at least one global parity,and d is
 6. 20. The array of storage devices of claim 16, wherein theplurality of storage devices are linked by two of: the first globalparity, the second global parity and the third global parity, and d isat least 7.